Exponential stability analysis via
Lyapunov method is extended to the one-dimensional heat and wave equations
with time-varying delay in the boundary conditions.
The delay function is admitted to be time-varying
with an a priori given upper
bound on its derivative, which is less than $1$.
Sufficient and explicit
conditions are derived that guarantee the exponential stability.
Moreover the decay rate can be explicitly computed if the data are
given.